Monday, September 20, 2010

The Story of the Blue Eyed Islanders

One of the most perplexing problems I have ever encountered is the infamous “Blue Eyed Islanders” puzzle. The perplexing aspect is a seeming paradox I will get to shortly, but the nominal solution involves straightforward induction, and is relatively easy to understand. It is possible that the story also has some applicability to more realistic human societeies.

The puzzle may be stated as follows:

There is a village where everyone has either blue eyes or brown eyes. Each person can see what eye color everyone else has, but each does not know the color of his or her own eyes. It is strictly forbidden for any islander to tell anyone else what color he/she has. If on a given day one happens to find out the color of ones own eyes, one must hurl oneself into the volcano at dawn the next day.One day an outsider comes to visit, a person who is known to always tell the truth. This person asserts for all to hear that “some of the people in this village have blue eyes”. Within a certain number of days, all in the village have killed themselves.The relatively easy parts of the puzzle are "why", and "when" does this happen?

Assume first that there is only one blue eyed person. He does not know that he has blue eyes until the fateful day of the outsider’s statement, but he does know that all of his fellow islanders have brown eyes. He thinks it is possible that he has brown eyes himself. But after the statement that there is at least one blue eyed person, he knows that it must be he himself, and the next day he commits “volcanacide”. If he were to see any other blue-eyed persons, he would not do himself in on the first dawn. So when the remaining brown eyed people see him do the deed, they all know that he saw all brown eyes, and thus they all have brown eyes. On the second dawn, they too jump into the fiery pit.

It is easy to see that if there were two blue-eyed persons, say Sam and Sara, each would think it possible that the other blue eyed person is the only one. But Sam sees that when Sara does not jump into the volcano upon the first dawn, he knows that he also has blue eyes, and on the second dawn Sam jumps in, and so does Sara, for exactly the same reasoning with regard to Sam. All the rest would jump in the third day, since they then deduce that they all have brown eyes, since evidently only Sam and Sara had the blue eyes the outsider noticed.

From induction, it follows that for a village population of N people, with X blue-eyed people and N-X brown eyed people, all of the blue eyed people will hurl themselves into the volcano X days after the outsider’s assertion. On day X+1, the brown eyed people will follow.The paradox is that every individual person in the village already knew, before the outsider’s visit, that there were some people in the village that have blue eyes (assuming that there was more than 1). They simply could see them, because if you were a blue, you’d see X-1 blues, if you were a brown, you’d see X blues. So what did the outsider tell them that they did not already know? That is, what specific new information did the outsider give them that led to the mass suicide of the village?

To illustrate this more concretely: What if there were 1000 villagers, 400 (say) with blue eyes. Before the outsiders visit and his unfortunate faux pas, each native knew that there were at least 399 blue eyed people on the island. Doesn’t it seem absurd that the statement that there is at least one would cause all 400 of the blue eyed people to hurl themselves to their doom on the 400th dawn, and the rest of the browns would follow on the 401st dawn? Thus, in this example, it would take over a year for the village society to cease to exist.

It seems that if one accepts logical induction, as surely one must, this surprising outcome must occur. The situation was stable before the outsiders made his observation, but the observation rendered that society unstable, and it rather quickly became extinct. It makes one wonder about possible applicability to more realistic human societies: could it be that some of the seemingly arbitrary taboos that have been put in place in various closed civilizations could somehow serve to stabilize them? In the example above, the particular taboo seems silly and unrealistic (and have they no mirrors or reflecting surfaces?). And even if an arbitrary taboo does bring about stability, we must also ask whether it involves a morally acceptable restriction, and if it is the only way that such stability could be achieved.

Perhaps the morale to this story is that there should be no taboos, but rather that human societies should be based on universal morality, that in turn is based on game theoretic reasoning such as is involved in the “prisoners dilemma” (this is really a sort of a generalized “golden rule”, wherein each person respects the rights of others to “live and let live”). But that is another tale, hopefully to come in a later blog.

Monday, September 6, 2010

Regarding putting a Muslim building close to the Twin Towers site

This has created quite an outcry from people who should know better. I simply do not get it. It was not Islam per se that did the terrible 9/11 crime, but rather a small number of Muslims. How anyone can want to pin a crime committed by a few members of a religion on all of their members just boggles the mind. Surely nobody would object to Catholics building a church close to where some slaughter of Belfast Protestants by Catholics (or vice versa) took place.

I often hear people say that they do not dispute the right of the Muslims to build there, they just think it is “insensitive”, or “in bad taste”. Why is that? Again, though it seems so obvious it shouldn’t even have to be said, most Muslims are not guilty of the attack on the World Trade Center. And while it is true that the United States is currently at war with two predominantly Muslim countries, we are of course not at war with the Islamic religion.

If the proposed mosque (or Islamic community center) were known to be an al Qaeda cell, or any group claiming to be responsible for the WTC attack, then the issue would be different (although one might feel even then that there is an advantage to keeping ones enemies in a known place where they can be more easily watched).

Assuming the property is owned and the project is funded by a Muslim institution, there are no conceivable grounds for preventing them, or even discouraging them, from building it. We should even be proud of having them from putting it there, as it is a clear demonstration of our basic rights to freedom of expression and freedom of religion, as well as the recognition of property rights. Maybe it will even serve as a good example to the many religiously dogmatic Muslim states.

Political opinion: gay marriage and marijuana

I don’t usually like to ponder politics much, but I cannot help this one. How could that California Proposition 8, about banning gay marriage, even get on a ballot? I do not understand how that could happen. Anything involving the basic rights of US citizens should not be decided by majority vote. This is why we have a republic, not a strict democracy. The rights of any minority group are protected from being violated by majority sentiments. The judge that ruled Proposition 8 unconstitutional was definitely right.
I see in the various opinion forums and OpEd pages certain patterns in why some people oppose gay marriage.
The often used “slippery slope” argument, that allowing gay marriages opens the door to marriage among multiple individuals, to animals, to underage children, etc is ridiculous. Allowing gay marriage is better likened to repealing the laws against interracial marriage (some of which existed into the 1960’s I believe); that is, allowing individuals the freedom to pursue happiness and live and love the person of their choice.
People who counter that proposition 8 did allow civil unions between gay couples miss the point. There are financial advantages that are associated with the government recognizing the marriage relationship—to give just a single example, and probably far from the most important one in terms of the amount of money involved, the “married, filing jointly” option is an advantage in most cases in filing federal income taxes. And if such advantages are there for straight couples, it is wrong (discriminatory) to deny them to gay couples.
Further, anyone who can muster up much energy and enthusiasm for objecting to gay marriage is not only being wrongheaded, but also needs to “get a hobby”. There are real things out there to worry about, folks, and this is not one of them. One should “live and let live”.
One frequently hears or reads the argument that gay marriage will undermine the traditional marriage. Well, so what? The divorce rate associated with traditional marriage is very high, so I would say it has already been undermined, and in fact the institution of marriage is often hypocritical (a vow “Till death do us part” is now apparently uttered without conviction in many cases). But that said, I do not even see how gay marriage would even affect most married people. Are these people thinking that there will be some massive rush for heterosexuals to divorce, and marry members of their own sex?
I saw one letter to the editor in the local paper today that asserted that government should be concerned with stopping anything that “weakens the moral fiber of the nation”---or some such nonsense. Has this person never read anything about Nazi Germany? I have little doubt that that government would have considered that to be within its scope. No, worrying about something as nebulous as “the moral fiber of the nation” is most certainly not the role of the government. The State, according to the philosophy of the “founding fathers” and the US constitution, is limited to securing and protecting the rights of the individual. I have to add that there is something really revolting about denying an individual the right of an association with a loved one for some dubious and vague purpose such as the “moral fiber of the nation”.
Here is a practical reason why everyone should support gay marriage: given that males tend to be polygamous, whether heterosexual or homosexual, the institution of marriage has a tendency to minimize (not eliminate, of course) promiscuity. So, if marriage is available to gay male couples, the incidence of sexually transmitted diseases would be very likely to decrease in frequency.
Many of the letters to the editors of the media seem to confuse the strictures of their religious beliefs with what should be binding legally. But no!---remember the separation of church and state principle---you must differentiate between what you believe God approves of, and what the US government allows. This seems so obvious to me that I really do not have the energy to elaborate on it much.

And what’s up with this widespread stupidity in Colorado of trying to get the medical marijuana dispensaries shut down? What hypocrisy, since my suspicion is that many of these objectors use alcohol, tobacco, coffee, etc. I do not have the facts at my fingertips, but am certain that from a practical standpoint—which by the way is not the only, or even the main, consideration here---that liquor causes much more socially negative behavior and damage than marijuana does. It is odd---I would have thought that as the generation older than baby boomers begins to fade (I hate to say, “die off”), that people would come to their senses and stop the idiotic “war on drugs”. Of course, I believe the first order principle involved here is that it is not the business of the government to tell us what we can and what we cannot imbibe (of course, it should be a crime to drive on public roads while under the influence of alcohol or drugs).
Wouldn’t a great many of the human caused ills in the world be greatly mitigated if people would learn to just mind their own business, and to take a “live and let live” attitude. But, it seems that very hard for us humans to do.